Transcript Probability and Statistical Distributions

Sampling Distributions (§4.11 - 4.12)
• Typically we select sample data from a population in
order to compute some statistic of interest.
• If we were to take two random samples from the same
population, it would be very unlikely that we would find
that we have computed the exact same value of the
statistics.
• Hence, the value of the statistic will vary from sample
to sample. That is the statistic itself is a random
variable.
• In this lecture we discuss how statistics (functions of
data) have distributions of their own, and how those
distributions can be determined in some cases by
means of the Central Limit Theorem.
Sampling Dist-1
Sampling Distribution of the Mean
1. Because no one sample is exactly like the next, the
sample mean will vary from sample to sample, and
hence is itself a random variable.
2. Random variables have distributions, and since the
sample mean is a random variable it must have a
distribution.
3. Regardless of the distribution of the measurements
(discrete or continuous), the distribution of the
sample mean can be approximated by a normal
distribution [Central Limit Theorem].
4. If the sample mean has a normal distribution, we can
compute probabilities for specific events using the
properties of the normal distribution.
Sampling Dist-2
A Resampling Experiment
Suppose we are interested in finding the mean of a large population
of individuals. A population too large to census. We decided to take a
sample of 5 individuals, measure their responses and compute the
sample mean. Now suppose we did this 1000 times (I.e. generate
1000 samples of size 5 and hence 1000 means). What would the
distribution of these means look like?
Area under
curve is one.
To make things interesting,
assume the probability
density function of the
measurements in the
populations has an
exponential shape, with
mean 1.
Sampling Dist-3
A sampling experiment:
Draw a sample of size n
from any population.
Compute the sample mean
Add the sample mean to a list.
Do many times.
Construct Histogram/Frequency Table
Sampling Dist-4
Example Continued
1. Draw a random sample of 5 individuals
from population.
2. Compute sample mean.
3. Add mean to list.
4. If number of simulations less than 1000,
return to 1 else go to 5.
5. Make histogram of 1000 means.
Sample
1
2
3
4
5
Mean
1.3498
0.6293
0.7390
0.7377
1.2206
Sampling Dist-5
Mean of 10 Observations
1. Draw a random sample of 10
individuals from population.
2. Compute sample mean.
3. Add mean to list.
4. If number of simulations less than
1000, return to 1 else go to 5.
5. Make histogram of 1000 means.
Sampling Dist-6
Comparison
Note: vertical scales are different.
Means of 5
•
•
Means of 10
Both have mean of about 1.0.
Spread of right plot is narrower than left plot. Somehow, when we
look at the distribution of samples of size 10 we have less spread
Dist-7
than if we look at samples of size 5. Is there a generalSampling
rule here?
Central Limit Theorem of Statistics
If random samples, each with n measurements,
are repeatedly drawn from the same population
having true mean m and standard deviation s,
then when n is large, the relative frequency
histogram for the sample means (calculated from
the repeated samples) will be approximately
normal (bell-shaped) with mean m and standard
deviation s/n, that is,
X n ~ approxN(m,s
n)
Note: In addition, the approximation becomes closer to
true normal as n increases.
Sampling Dist-8
Population and Sampling
Distribution
Distribution of means of random
samples of size 10 from
population.
Distribution of
measurements in
population
X
Random
Sample
Mean:
m
Standard Deviation: s
Distribution:
Anything
Mean:
m
Standard Deviation: s/10
Distribution:
Approx
Normal
Sampling Dist-9
Illustrating the CLT for the distribution of the sample mean when drawing
samples from an exponential distribution of mean 1 (based on 1000 draws).
Sampling Dist-10
Standard Error of the Mean
The quantity s is referred to as the standard
deviation. It is a measure of spread in the
population.
The quantity s/n is referred to as the standard
error of the mean. It is a measure of spread in the
distribution of means of random samples of size n
from a population of measurements having true
standard deviation s. I.e. it is just the standard
deviation of X n .
Sampling Dist-11
Uses of the Central Limit Theorem
•
•
•
The Central Limit Theorem is “central” to Statistics
because it allows us to make inferences (decisions)
about unknown population parameters, from sample
estimates (statistics).
We can estimate the true mean and standard deviation
of a population using the sample mean and sample
standard deviation.
Using the sample mean, the sample standard
deviation, and the central limit theorem, we can
develop hypothesis tests to determine whether the
TRUE population mean is equal to some specific value,
AND/OR, construct confidence intervals for the true
mean.
Sampling Dist-12
Population: mean μ, std. dev. σ
Draw sample (size n)
Estimate:
m  x, s  s
Draw Inferences:
Quantify uncertainty in estimates (confidence
intervals and hypothesis tests).
Sampling Dist-13